How many Numbers puzzles are there in total?

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Gavin Chipper
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Re: How many Numbers puzzles are there in total?

Post by Gavin Chipper » Mon Jan 11, 2016 5:03 pm

Jon Corby wrote:
Graeme Cole wrote:Which target is solvable by the fewest selections?
947, but even then there 9,017 selections which solve and only 4,226 which do not.
It's then small steps to the next worst, 941 (9045 solve, 4,198 don't), and similarly onto 967, 933 and 937 (T-Cap's other guess of 911 is #23 in the list.)

The first 'baddun' that isn't in the 900s is 853 at rank #13 (9453 solve, 3790 don't)
There was a whole thread on this and no-one could be arsed back then.

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Re: How many Numbers puzzles are there in total?

Post by Gavin Chipper » Mon Jan 11, 2016 5:11 pm

There's also some interesting stuff here. Apparently 100, 75, 50, 25, 8, 9 is the best 4-large selection in terms of the most solvable targets. And 100, 2, 3, 5, 8, 9 is the best selection at solving consecutive targets up from 101 if there was no 999 limit. It can solve up to 1912.

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Re: How many Numbers puzzles are there in total?

Post by JimBentley » Mon Jan 11, 2016 5:14 pm

Jon Corby wrote:Here is the breakdown by type, and how far away you can get:

Code: Select all

  Zero away         1 away   2 away   3 away   4 away   5 away  6 away  7 away  8 away  9 away  10 away  11+ away
6S 1963726 (76.64%) 353472    65417   26954     15232     9859    7357    5780    4589    3945    3326     102493
1L 4966076 (95.11%) 220295    21730    5813      2527     1272     713     470     339     264     217       1676
2L 3192103 (96.23%)  12258     8487    1880       865      435     313     207     144     114      78        426
3L  693131 (91.79%)  53875     4577    1332       724      461     296     206     139      87      64        268
4L   43710 (88.40%)   4661      556     179       112       75      42      28      22      18      16         26
This is excellent work by the way Jon, and the more detail the better. Also can we have 'likes' back?

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Re: How many Numbers puzzles are there in total?

Post by Gavin Chipper » Mon Jan 11, 2016 5:18 pm

JimBentley wrote:
Jon Corby wrote:Here is the breakdown by type, and how far away you can get:

Code: Select all

  Zero away         1 away   2 away   3 away   4 away   5 away  6 away  7 away  8 away  9 away  10 away  11+ away
6S 1963726 (76.64%) 353472    65417   26954     15232     9859    7357    5780    4589    3945    3326     102493
1L 4966076 (95.11%) 220295    21730    5813      2527     1272     713     470     339     264     217       1676
2L 3192103 (96.23%)  12258     8487    1880       865      435     313     207     144     114      78        426
3L  693131 (91.79%)  53875     4577    1332       724      461     296     206     139      87      64        268
4L   43710 (88.40%)   4661      556     179       112       75      42      28      22      18      16         26
This is excellent work by the way Jon, and the more detail the better. Also can we have 'likes' back?
Yeah, I have to grudgingly admit he's doing some good work here.

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Re: How many Numbers puzzles are there in total?

Post by JimBentley » Mon Jan 11, 2016 5:24 pm

Gavin Chipper wrote:Yeah, I have to grudgingly admit he's doing some good work here.
You two aren't going to have a fight at COLIN are you? I for one am hoping you're not, I think it might spoil the atmosphere a bit.

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Re: How many Numbers puzzles are there in total?

Post by Jon Corby » Mon Jan 11, 2016 6:17 pm

I do need to add some "weighting" to each selection as well though, because I'm treating each one equally at the moment (which is right for some measures but not others).

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Re: How many Numbers puzzles are there in total?

Post by Fred Mumford » Mon Jan 11, 2016 9:12 pm

Which impossible numbers games by category are the furthest from being solvable?

I worded that terribly, but if I give 1 1 2 2 3 3 => 999 as the 6 small one, and presumably 25 1 1 2 2 3 => 999 as the 1 large one then you get the idea. Hopefully it gets more interesting with 2 large and above.

Thanks in advance.

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Re: How many Numbers puzzles are there in total?

Post by Jon Corby » Tue Jan 12, 2016 9:28 am

Fred Mumford wrote:Which impossible numbers games by category are the furthest from being solvable?

I worded that terribly, but if I give 1 1 2 2 3 3 => 999 as the 6 small one, and presumably 25 1 1 2 2 3 => 999 as the 1 large one then you get the idea. Hopefully it gets more interesting with 2 large and above.

Thanks in advance.
With 2 large the worst are 25 away:

25,50,1,1,2,2 target 875
25,100,1,1,2,2 target 950
50,75,1,1,2,2 target 850
75,100,1,1,2,2 target 925

With 3 large the worst is 75 away:

50,75,100,1,1,2 target 975

With 4 large the worst is just 16 away:

25,50,75,100,1,1 target 866

Not sure if that was what you were hoping for in terms of being more interesting - the small numbers are fairly predictable in each case, but at least there's a bit of variation with the large ones.

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Re: How many Numbers puzzles are there in total?

Post by Jon Corby » Tue Jan 12, 2016 11:10 am

By the way, here is the correctly weighted distribution of games:

Code: Select all

     0 away   1 away   2 away   3 away   4 away   5 away   6 away   7 away   8 away   9 away   10 away   11+ away
6S   83.87%   10.48%    1.69%    0.68%    0.38%    0.24%    0.18%    0.14%    0.11%    0.09%     0.07%      2.08%
1L   97.75%    2.04%    0.14%    0.03%    0.01%    0.01%    0.00%    0.00%    0.00%    0.00%     0.00%      0.01%
2L   98.19%    1.69%    0.09%    0.02%    0.01%    0.00%    0.00%    0.00%    0.00%    0.00%     0.00%      0.00%
3L   94.73%    4.83%    0.27%    0.07%    0.04%    0.02%    0.01%    0.01%    0.01%    0.00%     0.00%      0.01%
4L   90.45%    8.23%    0.77%    0.24%    0.13%    0.07%    0.03%    0.02%    0.01%    0.01%     0.01%      0.02%
(NB, rounded to two decimal places, even where it says 0.00% there is still a value there, it's just very small! There are no proper zeroes in the grid.)

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Re: How many Numbers puzzles are there in total?

Post by Jon Corby » Tue Jan 12, 2016 1:48 pm

JimBentley wrote:961 is also a significant number and I will give a prize of one new penny for the first person (who isn't Jon) to say why.
Clue?

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Re: How many Numbers puzzles are there in total?

Post by Jon Corby » Tue Jan 12, 2016 2:15 pm

If you have small numbers 6,7,8,9 and it's either a 2 large, or a 1 large with the 100, TEN POINTS ARE AVAILABLE.

Struggling for a methodical way to look for commonalities like this though, there's probably more interesting (i.e. less specific) ones. Any ideas?

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Re: How many Numbers puzzles are there in total?

Post by JimBentley » Tue Jan 12, 2016 2:59 pm

Jon Corby wrote:
JimBentley wrote:961 is also a significant number and I will give a prize of one new penny for the first person (who isn't Jon) to say why.
Clue?
It's difficult to give a clue without giving it away. Actually, your figures are likely more accurate anyway so I might as well tell you what I think the answer is and ask you if it's right - I think it's the number with the "most complicated" solutions, if that makes sense? i.e. the target that - when you can get it spot-on - requires the most numbers to solve, usually all six.

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Re: How many Numbers puzzles are there in total?

Post by Jon Corby » Tue Jan 12, 2016 3:20 pm

JimBentley wrote:
Jon Corby wrote:
JimBentley wrote:961 is also a significant number and I will give a prize of one new penny for the first person (who isn't Jon) to say why.
Clue?
It's difficult to give a clue without giving it away. Actually, your figures are likely more accurate anyway so I might as well tell you what I think the answer is and ask you if it's right - I think it's the number with the "most complicated" solutions, if that makes sense? i.e. the target that - when you can get it spot-on - requires the most numbers to solve, usually all six.
Ah, I did wonder if it was anything along those lines; my data does not have any details about the solution, so I can neither confirm nor deny. My first run was writing out "the quickest solution" (defined as fewest steps, and lowest values of operands), but this was going to take about a month to run. By binning that I was able to take it down to about a day. I didn't really think there was THAT much worth in keeping the methods themselves, since there's nothing really set-based you can do with them, unless you were to keep EVERY different method I suppose, then you could look for commonalities. You're more likely to be curious about an individual game's method, which you can generate in a second anyway. A second for each of the near-12m games is about three months though...

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Re: How many Numbers puzzles are there in total?

Post by Dave Ricesky » Tue Jan 12, 2016 4:11 pm

Jon, I think you should do a similar analysis on the solvability of numbers games with the "nasty" large numbers as used in a couple of special episodes (those larges are 12, 37, 62, 87). I've done that analysis in the past and there's something very obviously surprising about the results when compared with the standard larges...

Answer: Picking 3 large leads to an absurdly high percentage of solvable games

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Re: How many Numbers puzzles are there in total?

Post by Jon Corby » Tue Jan 12, 2016 4:37 pm

Dave Ricesky wrote:Jon, I think you should do a similar analysis on the solvability of numbers games with the "nasty" large numbers as used in a couple of special episodes (those larges are 12, 37, 62, 87).
Since you asked so nicely, you're on! Running now, so hopefully should have some results tomorrow...

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Re: How many Numbers puzzles are there in total?

Post by Fred Mumford » Tue Jan 12, 2016 9:42 pm

Jon Corby wrote:Not sure if that was what you were hoping for in terms of being more interesting - the small numbers are fairly predictable in each case, but at least there's a bit of variation with the large ones.
Cheers Jon. The 4 large one is interesting (a relative term of course) - it shows once again that 4 large is not nearly as stifling as the average viewer might assume. Getting a double 1 on the other hand very much is.

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Re: How many Numbers puzzles are there in total?

Post by Graeme Cole » Tue Jan 12, 2016 10:40 pm

Fred Mumford wrote:
Jon Corby wrote:Not sure if that was what you were hoping for in terms of being more interesting - the small numbers are fairly predictable in each case, but at least there's a bit of variation with the large ones.
Cheers Jon. The 4 large one is interesting (a relative term of course) - it shows once again that 4 large is not nearly as stifling as the average viewer might assume. Getting a double 1 on the other hand very much is.
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Re: How many Numbers puzzles are there in total?

Post by Zarte Siempre » Wed Jan 13, 2016 12:55 am

Graeme Cole wrote:
Fred Mumford wrote:
Jon Corby wrote:Not sure if that was what you were hoping for in terms of being more interesting - the small numbers are fairly predictable in each case, but at least there's a bit of variation with the large ones.
Cheers Jon. The 4 large one is interesting (a relative term of course) - it shows once again that 4 large is not nearly as stifling as the average viewer might assume. Getting a double 1 on the other hand very much is.
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Re: How many Numbers puzzles are there in total?

Post by Jon Corby » Wed Jan 13, 2016 11:31 am

Dave Ricesky wrote:Jon, I think you should do a similar analysis on the solvability of numbers games with the "nasty" large numbers as used in a couple of special episodes (those larges are 12, 37, 62, 87).
Cheers Dave for the suggestion, this looks like it might be pretty interesting.

Staggeringly, there is NO CHANGE WHATSOEVER in the figures for 6 small. They're all completely identical. I simply can't get my head around why this could be, but nonetheless it means we can ignore them for now.

I haven't delved below the surface yet, but I too am initially surprised how advantageous (in purely computational terms) the 'nasty' big numbers are. Initially I would've thought the lack of divisibility would be detrimental, but I guess I overestimated its usefulness versus (presumably) the fact that the numbers (overall) are smaller and therefore better for multiplying to (near) the targets. You can almost think of 12 as almost an honorary small number rather than a large, and I guess that having 12 as the 1 large is responsible for pulling the overall 1L figures down.

Here is the count of distinct games:

Code: Select all

      0 away            1 away   2 away   3 away   4 away   5 away   6 away   7 away   8 away   9 away   10 away   11+ away
1L   4906599 (93.97%)   258757    27733     8926     4218     2439     1503      935      714      578       491       8499
2L   3257138 (98.19%)    54641     3531      885      353      184      133       90       69       53        40        193
3L    748401 (99.10%)     6178      397      103       49       12        5        4        2        2         2          5
4L     48944 (98.99%)      437       51       10        3        0        0        0        0        0         0          0
And here is the weighted distribution taking into account how likely each game is:

Code: Select all

Type   0 away    1 away    2 away    3 away    4 away    5 away    6 away    7 away    8 away    9 away    10 away    11+ away
1L     96.85%     2.74%     0.22%     0.07%     0.03%     0.02%     0.01%     0.01%     0.00%     0.00%      0.00%       0.05%
2L     99.22%     0.73%     0.03%     0.01%     0.00%     0.00%     0.00%     0.00%     0.00%     0.00%      0.00%       0.00%
3L     99.59%     0.38%     0.02%     0.01%     0.00%     0.00%     0.00%     0.00%     0.00%     0.00%      0.00%       0.00%
4L     99.55%     0.41%     0.03%     0.01%     0.00%      -         -         -         -         -          -           -
Note that there are genuine zeroes in the second grid (rather than just being really small numbers that round to 0.00) so these are represented with a dash.

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Re: How many Numbers puzzles are there in total?

Post by Dave Ricesky » Wed Jan 13, 2016 11:57 am

Yep, that agrees with the rough memory I had from ~2 years ago when I did this myself. I don't remember 2 or 4 large being quite so similar to 3 large in terms of gettable targets, but my overall surprise at Nasty 3 large being THE most gettable numbers variant probably just dulled my memory of the others.

Thanks for the figures!

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Re: How many Numbers puzzles are there in total?

Post by sean d » Wed Jan 13, 2016 2:16 pm

I too am amazed by the Nasty 3L revelation! I guess, as Jon suggests, the smaller large numbers are more flexible. For starters multplying any two of 25, 50, 75 and 100 immediately catapults you way out of potential target range, where 12 x 37 = 444 or 12 x 62 = 744 are immediately usable. Experience and the stats suggest that high targets a good distance from a multiple of 25 are tough with 3 or 4 large, but I'd not have expected that a more rnmdom less interactive looking selection would yield better results.

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Re: How many Numbers puzzles are there in total?

Post by Matthew Tassier » Wed Jan 13, 2016 5:02 pm

25,50,75,100 are a very inefficient set of large numbers because their co-factors lead to a lot of duplication in achievable targets if you have 2+ large numbers. If you take this into account then 12,37,62,87 being more useful shouldn't be too big a surprise.

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Re: How many Numbers puzzles are there in total?

Post by Dave Ricesky » Wed Jan 13, 2016 5:11 pm

Jon Corby wrote:Since you asked so nicely, you're on! Running now, so hopefully should have some results tomorrow...
Jon, how long does your program take to solve all the numbers games? I've dug up my old code (and made a couple of tweaks) and it can chow through all the standard numbers games, including recording an optimal* solution to each one, in a matter of minutes. I'm happy to use it to answer any nagging questions people might have that haven't yet been answered.

Currently, I'm taking a look at what the best numbers are for 4L - i.e. which set of four large numbers produces the largest volume of solvable games in 4L selections. So far, the best set is 11, 13, 18, 57, with a mere 8 games where the target can't be reached exactly out of a possible 49445. If I used a more structured / cleverer search, I might be able to improve on that figure...

*optimal here means "is the shortest solution to write down"
Matthew Tassier wrote:25,50,75,100 are a very inefficient set of large numbers because their co-factors lead to a lot of duplication in achievable targets if you have 2+ large numbers. If you take this into account then 12,37,62,87 being more useful shouldn't be too big a surprise.
Great point, Matthew!

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Re: How many Numbers puzzles are there in total?

Post by Clive Brooker » Wed Jan 13, 2016 8:27 pm

Dave Ricesky wrote:I've dug up my old code (and made a couple of tweaks) and it can chow through all the standard numbers games, including recording an optimal* solution to each one, in a matter of minutes. I'm happy to use it to answer any nagging questions people might have that haven't yet been answered.
Here's one that you and/or Jon might like to think about. Every so often someone produces a solution which involves multiplying up to a very high number before dividing down again. Usually this turns out to be showboating either because the solution offered could've been sequenced differently or because there are other solutions which don't require the same gymnastics.

But sometimes these alternatives don't exist. What numbers solution requires the highest intermediate solution?

The question presupposes agreement on when an intermediate solution is required. Every solution is built up using a sequence of addition/subtraction and multiplication/division operations, and my thinking is that an intermediate solution is required when control transfers from one to the other. Something like that anyway. So if the target is 302 and you solve it with 100*75/25+2, you have one intermediate solution, 300. But if you do (100*75+50)/25, you have two, 7500 and 7550.

Whether a definition of intermediate solution along these lines leaves room for ambiguity I have no idea.

It would then be interesting to know what is the best anyone has managed to do on the show.

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Re: How many Numbers puzzles are there in total?

Post by Thomas Carey » Wed Jan 13, 2016 8:44 pm

What about something like 100*75+6-6/25? 7500 is your intermediate, but...
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Re: How many Numbers puzzles are there in total?

Post by Clive Brooker » Wed Jan 13, 2016 8:50 pm

Thomas Carey wrote:What about something like 100*75+6-6/25? 7500 is your intermediate, but...
Any decent solver would regard that as equivalent to 100*75/25.

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Re: How many Numbers puzzles are there in total?

Post by Fred Mumford » Wed Jan 13, 2016 9:12 pm

Clive Brooker wrote:What numbers solution requires the highest intermediate solution?
100 75 50 25 3 3 => 996 ?

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Re: How many Numbers puzzles are there in total?

Post by Dave Ricesky » Wed Jan 13, 2016 9:14 pm

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Re: How many Numbers puzzles are there in total?

Post by Dave Ricesky » Wed Jan 13, 2016 9:15 pm

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Last edited by Dave Ricesky on Wed Jan 13, 2016 9:16 pm, edited 1 time in total.

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Re: How many Numbers puzzles are there in total?

Post by Dave Ricesky » Wed Jan 13, 2016 9:15 pm

Clive Brooker wrote:But sometimes these alternatives don't exist. What numbers solution requires the highest intermediate solution?
I'm afraid that's one question I can't answer without significantly slowing down my solver - some optimisations I've made to it mean that it discards quite a lot of potential solutions early on to remove duplication, and the result is that it might find a showboating solution and completely miss an easier alternative. The only guarantee I can make about the final solution is that it will always be the one involving the fewest starting numbers.
Fred Mumford wrote:100 75 50 25 3 3 => 996 ?
We probably have a winner.

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Re: How many Numbers puzzles are there in total?

Post by JimBentley » Wed Jan 13, 2016 9:42 pm

Dave Ricesky wrote:
Clive Brooker wrote:But sometimes these alternatives don't exist. What numbers solution requires the highest intermediate solution?
I'm afraid that's one question I can't answer without significantly slowing down my solver - some optimisations I've made to it mean that it discards quite a lot of potential solutions early on to remove duplication, and the result is that it might find a showboating solution and completely miss an easier alternative. The only guarantee I can make about the final solution is that it will always be the one involving the fewest starting numbers.
Fred Mumford wrote:100 75 50 25 3 3 => 996 ?
We probably have a winner.
Numbers: 100 75 50 25 3 3
Target: 996

found solution:
50 + 3 = 53
53 * 25 = 1325
1325 + 3 = 1328
1328 * 75 = 99600
99600 / 100 = 996

number of operations attempted: 1685356
number of solutions: 1

There must be something better than that, surely?

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Re: How many Numbers puzzles are there in total?

Post by Clive Brooker » Wed Jan 13, 2016 10:20 pm

JimBentley wrote:
Dave Ricesky wrote: We probably have a winner.
Numbers: 100 75 50 25 3 3
Target: 996

found solution:
50 + 3 = 53
53 * 25 = 1325
1325 + 3 = 1328
1328 * 75 = 99600
99600 / 100 = 996

number of operations attempted: 1685356
number of solutions: 1

There must be something better than that, surely?
I would argue that there is no need to evaluate 1328*75. An intermediate solution is only required when a sequence of addition and subtraction operations ends and a sequence of multiplication and division operations begins, and vice-versa.

Having got to 1328, all you need to demonstrate is that 1328*75 is a multiple of 100, equivalent to showing that 1328 divides by 4.

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Re: How many Numbers puzzles are there in total?

Post by Gavin Chipper » Thu Jan 14, 2016 12:13 am

This this discussion here too about intermediate targets.

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Re: How many Numbers puzzles are there in total?

Post by Thomas Carey » Thu Jan 14, 2016 3:54 am

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Re: How many Numbers puzzles are there in total?

Post by Jon Corby » Thu Jan 14, 2016 1:07 pm

Dave Ricesky wrote:Jon, how long does your program take to solve all the numbers games? I've dug up my old code (and made a couple of tweaks) and it can chow through all the standard numbers games, including recording an optimal* solution to each one, in a matter of minutes.
That's mightily impressive - is there any intelligence to it? I originally wrote a "crude" solver which just brute-forced every combination of sums for a VB Countdown game I wrote years ago, and didn't bother to finesse it much because it easily solved within a second or two, so there was no value in improving it. I converted the essence of it to c# (purely because that's what I program at work) to do this, but made a few tweaks to speed it up (notably 'caring' more about the numbers themselves - don't repeat sums with the same number if there's more than one of them, don't divide by 1, don't subtract if the larger number is double the smaller, don't multiply by 1.... think that's it) and was pleased enough to speed it up so that it did the circa 12m in about 14 hours! How on earth do you get it so quick?

Oh, another thing I suppose is that I pretty much "forget" each game and treat each one anew*, so I did wonder if there could be any value from "learning" from solves already done. But my hunch there was that the overheads if doing that lookup would outweigh the benefits.

* the one exception being I take a selection and solve the targets in ascending order; if the solve for target n is m where m > n, I don't need to do any more solves until n > m. This probably would have worked better in descending order actually, especially for those shit 6 smalls where you can only reach small numbers anyway... but again, it wouldn't reduce it to minutes instead of hours!!

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Re: How many Numbers puzzles are there in total?

Post by sean d » Thu Jan 14, 2016 1:20 pm

Clive Brooker wrote:
JimBentley wrote:
Dave Ricesky wrote: We probably have a winner.
Numbers: 100 75 50 25 3 3
Target: 996

found solution:
50 + 3 = 53
53 * 25 = 1325
1325 + 3 = 1328
1328 * 75 = 99600
99600 / 100 = 996

number of operations attempted: 1685356
number of solutions: 1

There must be something better than that, surely?
I would argue that there is no need to evaluate 1328*75. An intermediate solution is only required when a sequence of addition and subtraction operations ends and a sequence of multiplication and division operations begins, and vice-versa.

Having got to 1328, all you need to demonstrate is that 1328*75 is a multiple of 100, equivalent to showing that 1328 divides by 4.
The point with this one though Clive is that you can only use integers on the show (and on Apterous), so it is necessary to multiply by 75 before dividing by 100. According to Countdown rules it's unsolvable without going to 99,600. Essentially the whole thing is a variant on the 937.5 trick, i.e. (25 x 50 x 75) / 100, with some messing about to adjust the final outcome.

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Re: How many Numbers puzzles are there in total?

Post by Dave Ricesky » Thu Jan 14, 2016 1:47 pm

Jon Corby wrote:
Dave Ricesky wrote:Jon, how long does your program take to solve all the numbers games? I've dug up my old code (and made a couple of tweaks) and it can chow through all the standard numbers games, including recording an optimal* solution to each one, in a matter of minutes.
That's mightily impressive - is there any intelligence to it? I originally wrote a "crude" solver which just brute-forced every combination of sums for a VB Countdown game I wrote years ago, and didn't bother to finesse it much because it easily solved within a second or two, so there was no value in improving it. I converted the essence of it to c# (purely because that's what I program at work) to do this, but made a few tweaks to speed it up (notably 'caring' more about the numbers themselves - don't repeat sums with the same number if there's more than one of them, don't divide by 1, don't subtract if the larger number is double the smaller, don't multiply by 1.... think that's it) and was pleased enough to speed it up so that it did the circa 12m in about 14 hours! How on earth do you get it so quick?

Oh, another thing I suppose is that I pretty much "forget" each game and treat each one anew*, so I did wonder if there could be any value from "learning" from solves already done. But my hunch there was that the overheads if doing that lookup would outweigh the benefits.

* the one exception being I take a selection and solve the targets in ascending order; if the solve for target n is m where m > n, I don't need to do any more solves until n > m. This probably would have worked better in descending order actually, especially for those shit 6 smalls where you can only reach small numbers anyway... but again, it wouldn't reduce it to minutes instead of hours!!
It looks like the main difference is that I don't care about the target at all. To generate my stats, I take a selection of starting numbers and generate all the values you can make from that selection - once the program has generated all the possible results from the starting numbers, I just go through all the possible targets, and for each of them, see what the closest thing I generated was. Speed-up there of a factor of approximately 900...

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Re: How many Numbers puzzles are there in total?

Post by Clive Brooker » Thu Jan 14, 2016 2:00 pm

sean d wrote:The point with this one though Clive is that you can only use integers on the show (and on Apterous), so it is necessary to multiply by 75 before dividing by 100. According to Countdown rules it's unsolvable without going to 99,600. Essentially the whole thing is a variant on the 937.5 trick, i.e. (25 x 50 x 75) / 100, with some messing about to adjust the final outcome.
You don't need to evaluate 1328*75 to prove it's an integer because it automatically is.

If I'd been given the sum 1328*75/100 to work out at school, I would've expected a rap on the knuckles (metaphorical or otherwise) if I'd begun by multiplying 1328 by 75. I can't remember whether the Apterous interface requires you to add/subtract/multiply/divide one number at a time, but certainly on the show you could say ..... = 1328, times 75 divided by 100, and Rachel would be very happy with that.

Apologies for leading this off-topic. I originally responded to Dave's invitation to ask any numbers game questions we might have.

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Re: How many Numbers puzzles are there in total?

Post by JimBentley » Thu Jan 14, 2016 2:24 pm

Clive Brooker wrote:certainly on the show you could say ..... = 1328, times 75 divided by 100, and Rachel would be very happy with that.
I'm sure she would, as it follows the rules (going to 99,600 and coming back down, even if Rachel decided to omit writing that step down). Now if you said ... = 1328, times (75/100) (or ¾) then she should disallow it, as only integers can be used in calculations, so far as I know.

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Re: How many Numbers puzzles are there in total?

Post by sean d » Thu Jan 14, 2016 3:37 pm

If you're in true showboating mode you can go to 750,000 and back to a solution, incidentally.

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Re: How many Numbers puzzles are there in total?

Post by Jon Corby » Thu Jan 14, 2016 3:57 pm

Dave Ricesky wrote:It looks like the main difference is that I don't care about the target at all. To generate my stats, I take a selection of starting numbers and generate all the values you can make from that selection - once the program has generated all the possible results from the starting numbers, I just go through all the possible targets, and for each of them, see what the closest thing I generated was. Speed-up there of a factor of approximately 900...
Christ, do I feel stupid now. I'm repeating the exact same calculations hundreds of times over.,, :oops:

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Re: How many Numbers puzzles are there in total?

Post by Charlie Reams » Sat Jan 16, 2016 9:49 pm

Great topic, enjoyed reading all this!

Harder question: if one were designing a new variant, which four large numbers (in the range 11-100) make the game hardest (i.e. fewest games solvable) and which easiest (i.e. most games solvable)?

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Re: How many Numbers puzzles are there in total?

Post by Dave Ricesky » Sat Jan 16, 2016 10:07 pm

Charlie, I'll be able to answer that question restricted only to 4L picks in the next few days. Over all picks 6S to 4L, it seems to me to be intractable to get a complete answer at the moment.

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Re: How many Numbers puzzles are there in total?

Post by Gavin Chipper » Sat Jan 16, 2016 10:21 pm

For the hardest, it might be something stupid like 11, 12, 13, 14 or even 100, 99, 98, 97.

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Re: How many Numbers puzzles are there in total?

Post by Clive Brooker » Sun Jan 17, 2016 10:49 am

If you want the hardest possible variant, what about choosing six from 1-100 with equal probability? You'd want two of each in the pack I would imagine. Or perhaps this already exists.

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Re: How many Numbers puzzles are there in total?

Post by Charlie Reams » Sun Jan 17, 2016 11:29 am

Gavin Chipper wrote:For the hardest, it might be something stupid like 11, 12, 13, 14 or even 100, 99, 98, 97.
Yeah, I think 97-100 is a good shout.
Dave Ricesky wrote: Charlie, I'll be able to answer that question restricted only to 4L picks in the next few days. Over all picks 6S to 4L, it seems to me to be intractable to get a complete answer at the moment.
I'm okay with an imperfect answer, I'm sure we can get somewhere with a bit of stochastic hill climbing or some such. (I haven't tried this myself yet but it would make an interesting contest I reckon.)

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Re: How many Numbers puzzles are there in total?

Post by Jon Corby » Sun Jan 17, 2016 12:02 pm

I'll get the dictionary.

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Re: How many Numbers puzzles are there in total?

Post by Dave Ricesky » Sun Jan 17, 2016 2:15 pm

Charlie Reams wrote:I'm okay with an imperfect answer, I'm sure we can get somewhere with a bit of stochastic hill climbing or some such. (I haven't tried this myself yet but it would make an interesting contest I reckon.)
Hill climbing methods don't seem directly applicable here, since everything in sight is discrete. I'll have a think about other way of approaching it, because you're right, it is interesting to see how well we can do with imperfect methods!

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Re: How many Numbers puzzles are there in total?

Post by Charlie Reams » Sun Jan 17, 2016 4:23 pm

Dave Ricesky wrote:
Charlie Reams wrote:I'm okay with an imperfect answer, I'm sure we can get somewhere with a bit of stochastic hill climbing or some such. (I haven't tried this myself yet but it would make an interesting contest I reckon.)
Hill climbing methods don't seem directly applicable here, since everything in sight is discrete. I'll have a think about other way of approaching it, because you're right, it is interesting to see how well we can do with imperfect methods!
Hill climbing still works in a discrete space doesn't it? You can't do gradient descent but you can still have neighbours and a quality metric and that's really all you need. I'm not convinced this will work all that well because changing one number by even a small amount could have a huge effect on the available solutions, but maybe it's somewhere to start.

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Re: How many Numbers puzzles are there in total?

Post by Dave Ricesky » Sun Jan 17, 2016 5:21 pm

Charlie Reams wrote:
Dave Ricesky wrote:
Charlie Reams wrote:I'm okay with an imperfect answer, I'm sure we can get somewhere with a bit of stochastic hill climbing or some such. (I haven't tried this myself yet but it would make an interesting contest I reckon.)
Hill climbing methods don't seem directly applicable here, since everything in sight is discrete. I'll have a think about other way of approaching it, because you're right, it is interesting to see how well we can do with imperfect methods!
Hill climbing still works in a discrete space doesn't it? You can't do gradient descent but you can still have neighbours and a quality metric and that's really all you need. I'm not convinced this will work all that well because changing one number by even a small amount could have a huge effect on the available solutions, but maybe it's somewhere to start.
You're right, of course. I meant something more along the lines of "horribly discontinuous" (as far as that makes sense in a discrete space) rather than "discrete" - things such as the spread of prime factors appearing in the page numbers (so that we get a wide range of possible multiples of them) are likely to be more important than roughly what the numbers are equal to.

Then again, I'm tempted to try hill climbing or some sort of genetic algorithm just to see what kinds of results we get. It's not the worst starting point...

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Re: How many Numbers puzzles are there in total?

Post by JimBentley » Sun Jan 17, 2016 6:54 pm

Jon Corby wrote:I'll get the dictionary.
I for one hope that this problem with prove soluble through the use of the backward Kolmogorov equation and Ehrenhaft games, possibly (or not) incorporating the use of Markov chains*.


* I have absolutely no idea what I'm on about

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Re: How many Numbers puzzles are there in total?

Post by Dave Ricesky » Sun Jan 17, 2016 7:29 pm

Dave Ricesky wrote:Charlie, I'll be able to answer that question restricted only to 4L picks in the next few days. Over all picks 6S to 4L, it seems to me to be intractable to get a complete answer at the moment.
Changed my mind. I can answer the whole thing within the next week, probably.

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Re: How many Numbers puzzles are there in total?

Post by Jon Corby » Sun Jan 17, 2016 7:47 pm

JimBentley wrote:
Jon Corby wrote:I'll get the dictionary.
I for one hope that this problem with prove soluble through the use of the backward Kolmogorov equation and Ehrenhaft games, possibly (or not) incorporating the use of Markov chains*.


* I have absolutely no idea what I'm on about
I think this is at least the second time I've tried this when somebody has used the word stochastic. Maybe one day...

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Re: How many Numbers puzzles are there in total?

Post by Gavin Chipper » Sun Jan 17, 2016 7:57 pm

Jon Corby wrote:
JimBentley wrote:
Jon Corby wrote:I'll get the dictionary.
I for one hope that this problem with prove soluble through the use of the backward Kolmogorov equation and Ehrenhaft games, possibly (or not) incorporating the use of Markov chains*.


* I have absolutely no idea what I'm on about
I think this is at least the second time I've tried this when somebody has used the word stochastic. Maybe one day...
But maybe not. It's unpredictable.

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Re: How many Numbers puzzles are there in total?

Post by JimBentley » Sun Jan 17, 2016 8:29 pm

Jon Corby wrote:
JimBentley wrote:
Jon Corby wrote:I'll get the dictionary.
I for one hope that this problem with prove soluble through the use of the backward Kolmogorov equation and Ehrenhaft games, possibly (or not) incorporating the use of Markov chains*.


* I have absolutely no idea what I'm on about
I think this is at least the second time I've tried this when somebody has used the word stochastic. Maybe one day...
Have you read "The Stochastic Man" as well, then?

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Re: How many Numbers puzzles are there in total?

Post by Graeme Cole » Sun Jan 17, 2016 9:30 pm

Jon Corby wrote:
JimBentley wrote:
Jon Corby wrote:I'll get the dictionary.
I for one hope that this problem with prove soluble through the use of the backward Kolmogorov equation and Ehrenhaft games, possibly (or not) incorporating the use of Markov chains*.


* I have absolutely no idea what I'm on about
I think this is at least the second time I've tried this when somebody has used the word stochastic. Maybe one day...
Are you referencing that Simpsons episode set in the future?

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Re: How many Numbers puzzles are there in total?

Post by Jon Corby » Mon Jan 18, 2016 9:12 am

Graeme Cole wrote:Are you referencing that Simpsons episode set in the future?
You'll see when you get there.

One of the best episodes fo sho.

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Re: How many Numbers puzzles are there in total?

Post by Dave Ricesky » Fri Jan 22, 2016 11:22 am

Just an update on Charlie's question - I've got full statistics for how many 0,1,2 or 3 large games are gettable with each possible set of large numbers. I was expecting to be completely done on the 4L by now as well, but my program hit a snag after I kicked it off and I didn't notice until now. For certain choices of large numbers, you can reach values larger than 2^31, causing overflow and other nasty things to happen to my program.

Looks like it got about 50% done with the 4L stuff before overflow became a problem. Rewritten using 64 bit integers and rerunning :)

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Re: How many Numbers puzzles are there in total?

Post by Ian Volante » Fri Jan 22, 2016 1:34 pm

Dave Ricesky wrote:you can reach values larger than 2^31
Really? How, given that 25*50*75*100*10*10 is under 1x10^9 and 2^31 is >2x10^9? Some intermediate factoring steps?
meles meles meles meles meles meles meles meles meles meles meles meles meles meles meles meles

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Re: How many Numbers puzzles are there in total?

Post by Dave Ricesky » Fri Jan 22, 2016 1:43 pm

Ian Volante wrote:
Dave Ricesky wrote:you can reach values larger than 2^31
Really? How, given that 25*50*75*100*10*10 is under 1x10^9 and 2^31 is >2x10^9? Some intermediate factoring steps?
Charlie's question was about using different sets of large numbers - and 100*99*98*97*10*10 is way too big, for example.

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